scipy.linalg.invpascal#
- scipy.linalg.invpascal(n, kind='symmetric', exact=True)[source]#
Returns the inverse of the n x n Pascal matrix.
The Pascal matrix is a matrix containing the binomial coefficients as its elements.
- Parameters
- nint
The size of the matrix to create; that is, the result is an n x n matrix.
- kindstr, optional
Must be one of ‘symmetric’, ‘lower’, or ‘upper’. Default is ‘symmetric’.
- exactbool, optional
If exact is True, the result is either an array of type
numpy.int64
(if n <= 35) or an object array of Python integers. If exact is False, the coefficients in the matrix are computed usingscipy.special.comb
with exact=False. The result will be a floating point array, and for large n, the values in the array will not be the exact coefficients.
- Returns
- invp(n, n) ndarray
The inverse of the Pascal matrix.
See also
Notes
New in version 0.16.0.
References
- 1
“Pascal matrix”, https://en.wikipedia.org/wiki/Pascal_matrix
- 2
Cohen, A. M., “The inverse of a Pascal matrix”, Mathematical Gazette, 59(408), pp. 111-112, 1975.
Examples
>>> from scipy.linalg import invpascal, pascal >>> invp = invpascal(5) >>> invp array([[ 5, -10, 10, -5, 1], [-10, 30, -35, 19, -4], [ 10, -35, 46, -27, 6], [ -5, 19, -27, 17, -4], [ 1, -4, 6, -4, 1]])
>>> p = pascal(5) >>> p.dot(invp) array([[ 1., 0., 0., 0., 0.], [ 0., 1., 0., 0., 0.], [ 0., 0., 1., 0., 0.], [ 0., 0., 0., 1., 0.], [ 0., 0., 0., 0., 1.]])
An example of the use of kind and exact:
>>> invpascal(5, kind='lower', exact=False) array([[ 1., -0., 0., -0., 0.], [-1., 1., -0., 0., -0.], [ 1., -2., 1., -0., 0.], [-1., 3., -3., 1., -0.], [ 1., -4., 6., -4., 1.]])